Tan 54 Degrees
The value of tan 54 degrees is 1.3763819. . .. Tan 54 degrees in radians is written as tan (54° × π/180°), i.e., tan (3π/10) or tan (0.942477. . .). In this article, we will discuss the methods to find the value of tan 54 degrees with examples.
- Tan 54°: √(25 + 10√5)/5
- Tan 54° in decimal: 1.3763819. . .
- Tan (-54 degrees): -1.3763819. . . or -√(25 + 10√5)/5
- Tan 54° in radians: tan (3π/10) or tan (0.9424777 . . .)
What is the Value of Tan 54 Degrees?
The value of tan 54 degrees in decimal is 1.376381920. . .. Tan 54 degrees can also be expressed using the equivalent of the given angle (54 degrees) in radians (0.94247 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 54 degrees = 54° × (π/180°) rad = 3π/10 or 0.9424 . . .
∴ tan 54° = tan(0.9424) = √(25 + 10√5)/5 or 1.3763819. . .
Explanation:
For tan 54 degrees, the angle 54° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 54° value = √(25 + 10√5)/5 or 1.3763819. . .
Since the tangent function is a periodic function, we can represent tan 54° as, tan 54 degrees = tan(54° + n × 180°), n ∈ Z.
⇒ tan 54° = tan 234° = tan 414°, and so on.
Note: Since, tangent is an odd function, the value of tan(-54°) = -tan(54°).
Methods to Find Value of Tan 54 Degrees
The tangent function is positive in the 1st quadrant. The value of tan 54° is given as 1.37638. . .. We can find the value of tan 54 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Tan 54 Degrees Using Unit Circle
To find the value of tan 54 degrees using the unit circle:
- Rotate ‘r’ anticlockwise to form 54° angle with the positive x-axis.
- The tan of 54 degrees equals the y-coordinate(0.809) divided by x-coordinate(0.5878) of the point of intersection (0.5878, 0.809) of unit circle and r.
Hence the value of tan 54° = y/x = 1.3764 (approx).
Tan 54° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 54 degrees as:
- sin(54°)/cos(54°)
- ± sin 54°/√(1 - sin²(54°))
- ± √(1 - cos²(54°))/cos 54°
- ± 1/√(cosec²(54°) - 1)
- ± √(sec²(54°) - 1)
- 1/cot 54°
Note: Since 54° lies in the 1st Quadrant, the final value of tan 54° will be positive.
We can use trigonometric identities to represent tan 54° as,
- cot(90° - 54°) = cot 36°
- -cot(90° + 54°) = -cot 144°
- -tan (180° - 54°) = -tan 126°
☛ Also Check:
FAQs on Tan 54 Degrees
What is Tan 54 Degrees?
Tan 54 degrees is the value of tangent trigonometric function for an angle equal to 54 degrees. The value of tan 54° is √(25 + 10√5)/5 or 1.3764 (approx).
What is the Value of Tan 54° in Terms of Sec 54°?
We can represent the tangent function in terms of the secant function using trig identities, tan 54° can be written as √(sec²(54°) - 1). Here, the value of sec 54° is equal to 1.7013.
What is the Value of Tan 54 Degrees in Terms of Sin 54°?
Using trigonometric identities, we can write tan 54° in terms of sin 54° as, tan(54°) = sin 54°/√(1 - sin²(54°)) . Here, the value of sin 54° is equal to (√5 + 1)/4.
How to Find Tan 54° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 54° can be given in terms of other trigonometric functions as:
- sin(54°)/cos(54°)
- ± sin 54°/√(1 - sin²(54°))
- ± √(1 - cos²(54°))/cos 54°
- ± 1/√(cosec²(54°) - 1)
- ± √(sec²(54°) - 1)
- 1/cot 54°
☛ Also check: trigonometric table
How to Find the Value of Tan 54 Degrees?
The value of tan 54 degrees can be calculated by constructing an angle of 54° with the x-axis, and then finding the coordinates of the corresponding point (0.5878, 0.809) on the unit circle. The value of tan 54° is equal to the y-coordinate(0.809) divided by the x-coordinate (0.5878). ∴ tan 54° = √(25 + 10√5)/5 or 1.3764